In case of type MI, the covering \(K3\)  surface \(X\)  is the Fermat quartic surface and the covering involution is induced from a pair of skew lines on \(X\) . Note that the defining equation of the Fermat quartic surface is a hermitian form over \({\bf F}_9\)  and hence the projective unitary group \({\rm PGU}_4({\bf F}_9)\)  acts on the set of pairs of two skew lines. Recall that
\(|{\rm PGU}_4({\bf F}_9)|= 2^9\cdot 3^6\cdot 5\cdot 7\)  (e.g. Atlas [1]}).
The number of pairs of skew lines is \(112\cdot 81 /2=2^3\cdot 3^4 \cdot 7\)  and the stabilizer subgroup of a pair of skew lines is \({\rm Aut}(S) \times {\bf Z}/2{\bf Z}\)  which has order \(2^6\cdot 3^2\cdot 5\) . Here \({\bf Z}/2{\bf Z}\)  is generated by the covering involution of the double covering \(X\rightarrow S\) .
Thus the action is transitive, and hence the uniqueness of \(S\)  follows.
